ShareButton

Friday, November 16, 2012

Fun with Uniform Random Numbers

Q: You have two uniformly random numbers x and y (meaning they can take any value between 0 and 1 with equal probability). What distribution does the sum of these two random numbers follow? What is the probability that their product is less than 0.5.

A: Let z = x + y be the random variable whose distribution we want. Clearly z runs from 0 to 2. Let 'f' denote the uniform random distribution between [0,1]. An important point to understand is that f has a fixed value of 1 when x runs from 0 to 1 and its 0 otherwise.

So the probability density for z, call it P(z) at any point is the product of f(y) and f(z-y), where y runs from 0 to 1.

However in that range f(y) is equal to 1. So the above equation becomes

From here on, it gets a bit tricky. Notice that the integral is a function of z. Let us take a look at how else we can simply the above integral.

It is easy to see that f(z-y) = 1 when (z-y) is between [0,1]. This is the same as saying

Likewise, f(z-y) = 1 when y is lesser than z and greater than 0. ie

Combining the two cases above results in a discontinuous function as

which is a triangular function.

Now that we done with the sum, what about the product xy? A quick way to go about it is to visualize a 2 dimensional plane. All the points (x,y) within the square [0,1]x[0,1] fall in the candidate space. The case when xy = 0.5 makes a curve

The area under the curve would represent the cases for which xy <= 0.5 (shown shaded below). Since the area for the square is 1, that area is the sought probability.

The curve intersects the square at [0.5,1] and [1,0.5]. The area under the curve would the be sum of the 2 quadrants (1/4 each) along with the integral of y = 0.5/x under the range 0.5 to 1 yielding
$$
\begin{align*}
P(xy\lt 0.5) &= \frac{1}{2} + \int_{0.5}^{1}\frac{0.5}{x}dx \\
&= \frac{1}{2} + \frac{1}{2}ln2 \approx 0.85
\end{align*}
$$

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability